Unstable Poles--Unit Circle Viewpoint
We saw in §8.4 that an LTI filter is stable if and only if all of its poles are strictly inside the unit circle () in the complex plane. In particular, a pole outside the unit circle () gives rise to an impulse-response component proportional to which grows exponentially over time . We also saw in §6.2 that the z transform of a growing exponential does not not converge on the unit circle in the plane. However, this was the case for a causal exponential , where is the unit-step function (which switches from 0 to 1 at time 0). If the same exponential is instead anticausal, i.e., of the form , then, as we'll see in this section, its z transform does exist on the unit circle, and the pole is in exactly the same place as in the causal case. Therefore,to unambiguously invert a z transform, we must know its region of convergence. The critical question is whether the region of convergence includes the unit circle: If it does, then each pole outside the unit circle corresponds to an anticausal, finite energy, exponential, while each pole inside corresponds to the usual causal decaying exponential.
The essence of the situation can be illustrated using a simple geometric series. Let be any real (or complex) number. Then we have
One-Pole Transfer Functions
We can apply the same analysis to a one-pole transfer function. Let denote any real or complex number:
Now consider the rewritten case:
where the inverse z transform is the inverse bilateral z transform. In this case, the convergence criterion is , or , and this region includes the unit circle when .
In summary, when the region-of-convergence of the z transform is assumed to include the unit circle of the plane, poles inside the unit circle correspond to stable, causal, decaying exponentials, while poles outside the unit circle correspond to anticausal exponentials that decay toward time , and stop before time zero.
Figure 8.8 illustrates the two types of exponentials (causal and anticausal) that correspond to poles (inside and outside the unit circle) when the z transform region of convergence is defined to include the unit circle.
myFourFiguresToWidthpolesout11polesout21polesout12polesout220.52Left column: Causal exponential decay, pole at . Right column: Anticausal exponential decay, pole at . Top: Pole-zero diagram. Bottom: Corresponding impulse response, assuming the region of convergence includes the unit circle in the plane.
Poles and Zeros of the Cepstrum
Time Constant of One Pole